Optical waveguides including multi-mode dielectric waveguides are designed for the transmission of electromagnetic waves in the optical band. An optical waveguide is basically a light conduit configured, by means of properly selected core and surrounding cladding materials with higher and lower refractive indexes, respectively, to confine and transport light therein without leaking it to the environment.
Optical waveguides can be classified according to their geometry (slab (planar) or strip, cylindrical, etc.), mode structure (single-mode, multi-mode), refractive index distribution (step or gradient index) and e.g. material (glass, polymer, semiconductor, etc.). Refraction of light at the core/cladding interface is generally governed by the Snell's law. When light arrives at the interface between the core and cladding materials above a so-called critical angle it is completely reflected back into the core material based on a phenomenon called ‘total internal reflection’ (TIR).
In terms of wave-optics, a multi-mode waveguide is, as the name alludes, capable of guiding the waves of several modes, i.e. a discrete set of solutions of Maxwell's equations with boundary conditions, in addition to the main mode. In practice, the larger the core dimensions of the waveguide the greater the number of modes is. The multi-mode waveguides and related equipment for interfacing the light with the waveguide are typically easier to construct than the single-mode counterparts due to e.g. larger dimensions generally enabling the utilization of coarser, more affordable hardware and manufacturing methods. However, multimode distortion limits the ‘bandwidth×distance’ product of multi-mode waveguides in contrast to single-mode solutions. It also complicates (or prevents) the realization of advanced waveguide circuits that densely integrate a large number of waveguide components (couplers, filters etc.). Therefore, such circuits are typically realized with single-mode waveguides, while multimode waveguides are mainly used in point-to-point links and to realize relatively simple waveguide circuits.
Multimode waveguides can also be locally used as part of single-mode waveguide circuits. They can form components, such as multi-mode interference (MMI) couplers, where multiple modes are temporarily excited, but the light eventually couples back into single-mode waveguides. They can also be used to propagate light only in the fundamental mode, but in this case light must be coupled adiabatically between the single and multimode waveguide sections to avoid the excitation of higher order modes. And finally, multimode waveguides can be placed behind single-mode waveguides when multimode distortion is no longer relevant, for example when coupling light into a large-area photodetector.
By definition, the modes of a multi-mode straight waveguide propagate unperturbed without mutual coupling, unless some perturbation occurs, such as a change in the waveguide shape. In particular, bends can induce significant coupling between the different modes such that in the straight section at the end of the bend, also higher order modes (HOM) will be in general excited, even if only the fundamental mode was excited in a straight section preceding the bend. The higher the curvature 1/R (bend radius R), the higher is the degree of unwanted coupling and, in general, also the higher the number of significantly excited modes.
Indeed, one basic design rule of single-moded photonic integrated circuits dictates that any bent waveguide must be single-moded so that the undesired coupling between the modes and subsequent detrimental mode beating and power radiation in the bend may be avoided. For integration purposes the bend radius is typically to be minimized, which requires the use of HIC waveguides. Further, the higher the index contrast the smaller the waveguide shall be in order to ensure the single-mode condition. Sub-micron waveguides could be utilized for achieving dense integration, but they pose many additional challenges, including polarization dependence and low coupling efficiency to optical fibre modes. Furthermore, for scalable production they require expensive state-of-the-art fabrication tools in order to resolve submicron features and are also very sensitive to nanometer-scale fabrication errors.
As a reference one may introduce a single-moded rib waveguide that can be realized on a silicon-on-insulator (SOI) wafer by dry etching the originally 4 μm thick Si layer down to approximately 2 μm thickness around an unetched 3.5 μm wide rib that forms the waveguide. Despite its large dimensions and high index contrast this waveguide is single-moded because the higher order modes radiate power away from the rib along the surrounding 2 μm thick Si slab. However, the slab also enables the fundamental mode to radiate power into the slab when the rib waveguide is bent. Therefore the minimum bending radius for such a rib waveguides is approximately 4 mm. To avoid the radiation losses of the fundamental mode in a bend the rib waveguide can be locally converted into a multimode strip waveguide or the etch depth can be locally increased around the bend [Reference: K. Solehmainen, T Aalto, I Dekker, H. Kapulainen, H. Harjanne and P. Heimala, “Development of multi-step processing in silicon-on-insulator for optical waveguide applications”, Journal of Optics A: Pure and Applied Optics, vol 8, pp. S455-S460 (2006)]. However, in practice this has led to the inevitable excitation of HOMs if the bending radius has been reduced by a factor of 10 or more with respect to the corresponding low-loss rib waveguide bend.
The goal of shrinking the bend radius of multimode HIC waveguides could be sought by a matched arc approach, which relies on matching the length of a circular bend to an integer multiple of beating lengths between the fundamental mode and the first higher order mode (HOM) of the bent waveguide to ensure that, at the end of the bend, only the fundamental mode will be excited despite the fact that HOMs have been excited during propagation in the bent section. Nevertheless, the obtained bending radii are still relatively large, in practice e.g. two orders of magnitude larger than the waveguide width, and in particular, manufacturing thereof is challenging due to very stringent tolerance requirements.